Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(47,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([24, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bm (of order \(15\), degree \(8\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −2.47281 | − | 1.10097i | 0.911643 | + | 2.80575i | 3.56441 | + | 3.95867i | 0.118847 | − | 1.13076i | 0.834714 | − | 7.94178i | −0.206171 | + | 0.0438230i | −2.78282 | − | 8.56465i | −4.61409 | + | 3.35233i | −1.53881 | + | 2.66530i |
47.2 | −2.46938 | − | 1.09944i | −0.161350 | − | 0.496583i | 3.55081 | + | 3.94357i | −0.0870451 | + | 0.828178i | −0.147529 | + | 1.40365i | 3.22548 | − | 0.685597i | −2.76198 | − | 8.50051i | 2.20649 | − | 1.60311i | 1.12548 | − | 1.94939i |
47.3 | −2.46644 | − | 1.09813i | 0.427841 | + | 1.31676i | 3.53917 | + | 3.93065i | −0.393604 | + | 3.74489i | 0.390728 | − | 3.71753i | −4.24199 | + | 0.901662i | −2.74419 | − | 8.44575i | 0.876244 | − | 0.636628i | 5.08318 | − | 8.80432i |
47.4 | −2.46005 | − | 1.09528i | −0.858220 | − | 2.64133i | 3.51393 | + | 3.90262i | −0.0532761 | + | 0.506889i | −0.781744 | + | 7.43780i | −1.23013 | + | 0.261473i | −2.70569 | − | 8.32727i | −3.81303 | + | 2.77033i | 0.686249 | − | 1.18862i |
47.5 | −2.23412 | − | 0.994694i | 0.0984508 | + | 0.303000i | 2.66361 | + | 2.95824i | 0.0562497 | − | 0.535181i | 0.0814419 | − | 0.774868i | 2.72396 | − | 0.578996i | −1.49685 | − | 4.60683i | 2.34493 | − | 1.70369i | −0.658010 | + | 1.13971i |
47.6 | −2.15087 | − | 0.957629i | −0.755329 | − | 2.32466i | 2.37093 | + | 2.63318i | 0.190967 | − | 1.81693i | −0.601551 | + | 5.72338i | −2.24928 | + | 0.478099i | −1.12284 | − | 3.45573i | −2.40649 | + | 1.74842i | −2.15069 | + | 3.72510i |
47.7 | −2.06087 | − | 0.917558i | −0.708558 | − | 2.18072i | 2.06701 | + | 2.29564i | −0.372272 | + | 3.54193i | −0.540690 | + | 5.14432i | 0.890549 | − | 0.189292i | −0.759220 | − | 2.33664i | −1.82643 | + | 1.32698i | 4.01713 | − | 6.95788i |
47.8 | −1.96443 | − | 0.874619i | 0.796648 | + | 2.45183i | 1.75575 | + | 1.94996i | 0.0324014 | − | 0.308279i | 0.579461 | − | 5.51320i | 1.16764 | − | 0.248189i | −0.414593 | − | 1.27599i | −2.94977 | + | 2.14314i | −0.333277 | + | 0.577252i |
47.9 | −1.96204 | − | 0.873555i | −0.00351773 | − | 0.0108264i | 1.74823 | + | 1.94160i | 0.127016 | − | 1.20848i | −0.00255559 | + | 0.0243148i | −3.78448 | + | 0.804416i | −0.406628 | − | 1.25147i | 2.42695 | − | 1.76328i | −1.30488 | + | 2.26012i |
47.10 | −1.87859 | − | 0.836403i | 0.696822 | + | 2.14460i | 1.49128 | + | 1.65623i | 0.203556 | − | 1.93671i | 0.484704 | − | 4.61165i | −2.97503 | + | 0.632363i | −0.145313 | − | 0.447227i | −1.68669 | + | 1.22545i | −2.00227 | + | 3.46803i |
47.11 | −1.87605 | − | 0.835272i | −0.661083 | − | 2.03461i | 1.48363 | + | 1.64773i | 0.403601 | − | 3.84001i | −0.459222 | + | 4.36921i | 0.534911 | − | 0.113699i | −0.137863 | − | 0.424298i | −1.27554 | + | 0.926732i | −3.96463 | + | 6.86693i |
47.12 | −1.80828 | − | 0.805099i | 0.222654 | + | 0.685260i | 1.28344 | + | 1.42540i | −0.317759 | + | 3.02328i | 0.149080 | − | 1.41840i | −0.411179 | + | 0.0873988i | 0.0501149 | + | 0.154238i | 2.00705 | − | 1.45820i | 3.00864 | − | 5.21111i |
47.13 | −1.77387 | − | 0.789778i | −0.397407 | − | 1.22309i | 1.18460 | + | 1.31563i | 0.0674340 | − | 0.641592i | −0.261023 | + | 2.48347i | 2.84339 | − | 0.604382i | 0.137790 | + | 0.424075i | 1.08903 | − | 0.791226i | −0.626334 | + | 1.08484i |
47.14 | −1.70810 | − | 0.760495i | 0.501967 | + | 1.54490i | 1.00099 | + | 1.11171i | 0.450831 | − | 4.28937i | 0.317476 | − | 3.02058i | 4.54281 | − | 0.965603i | 0.291227 | + | 0.896305i | 0.292321 | − | 0.212383i | −4.03211 | + | 6.98382i |
47.15 | −1.48297 | − | 0.660260i | 0.874496 | + | 2.69142i | 0.424988 | + | 0.471997i | −0.286644 | + | 2.72723i | 0.480188 | − | 4.56868i | 1.00163 | − | 0.212902i | 0.684658 | + | 2.10716i | −4.05196 | + | 2.94392i | 2.22576 | − | 3.85513i |
47.16 | −1.43570 | − | 0.639214i | −0.970398 | − | 2.98658i | 0.314374 | + | 0.349148i | −0.114787 | + | 1.09213i | −0.515864 | + | 4.90812i | 3.78205 | − | 0.803899i | 0.743116 | + | 2.28708i | −5.55093 | + | 4.03298i | 0.862905 | − | 1.49460i |
47.17 | −1.34961 | − | 0.600883i | −0.579021 | − | 1.78204i | 0.122112 | + | 0.135620i | −0.163893 | + | 1.55934i | −0.289350 | + | 2.75298i | −4.75277 | + | 1.01023i | 0.829726 | + | 2.55363i | −0.413364 | + | 0.300327i | 1.15817 | − | 2.00601i |
47.18 | −1.32721 | − | 0.590914i | −0.369514 | − | 1.13725i | 0.0740567 | + | 0.0822483i | −0.328701 | + | 3.12738i | −0.181591 | + | 1.72772i | −0.648672 | + | 0.137879i | 0.848203 | + | 2.61050i | 1.27026 | − | 0.922898i | 2.28427 | − | 3.95647i |
47.19 | −1.08348 | − | 0.482395i | 0.207448 | + | 0.638460i | −0.397045 | − | 0.440964i | 0.0813463 | − | 0.773959i | 0.0832245 | − | 0.791828i | −1.78786 | + | 0.380021i | 0.950467 | + | 2.92524i | 2.06245 | − | 1.49846i | −0.461490 | + | 0.799325i |
47.20 | −1.04425 | − | 0.464931i | 0.562144 | + | 1.73010i | −0.463958 | − | 0.515278i | 0.00867475 | − | 0.0825347i | 0.217358 | − | 2.06802i | 2.17979 | − | 0.463328i | 0.951381 | + | 2.92805i | −0.250198 | + | 0.181779i | −0.0474316 | + | 0.0821539i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
61.c | even | 3 | 1 | inner |
671.bm | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bm.a | ✓ | 480 |
11.c | even | 5 | 1 | inner | 671.2.bm.a | ✓ | 480 |
61.c | even | 3 | 1 | inner | 671.2.bm.a | ✓ | 480 |
671.bm | even | 15 | 1 | inner | 671.2.bm.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bm.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
671.2.bm.a | ✓ | 480 | 11.c | even | 5 | 1 | inner |
671.2.bm.a | ✓ | 480 | 61.c | even | 3 | 1 | inner |
671.2.bm.a | ✓ | 480 | 671.bm | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).